Study of the Dynamic Characteristics of A Cone-Shaped Recovery System on Submarines for Recovering Autonomous Underwater Vehicle

National navies equip their submarines with Autonomous Underwater Vehicle (AUV) technology. It has become an important component of submarine development in technologically-advanced countries. Employing advanced and reliable recovery systems directly improves the safety and operational efficiency of submarines equipped with AUVs. In this paper, based on aerial refueling technology, a cone-shaped recovery system with two different guiding covers (closed structure and frame structure) is applied to the submarine. By taking the Suboff model as the research object, STAR-CCM was used to study the influence of the installation position of the recovery system, and the length of the rigid rod, on the Suboff model. It was found that when the recovery system is installed in the middle and rear of the Suboff model at the same velocity and the same length of the rigid rod, the Suboff model has the good stability and less drag. It experiences the largest drag when being installed in the front of the rigid rod. Moreover, when the recovery system is installed in the front and middle of the rigid rod, the drag increases as its length increases, and the lift decreases as its length increases. Compared with the closed structure guiding cover, the Suboff model will have less drag and better stability when the recovery system uses the frame structure guiding cover. Besides, the deflection and vibration of the rigid rod were also analyzed via mathematical theory.


Introduction
The saying "victory on land begins under the sea" emphasizes the importance of quality of underwater equipment in warfare. In 1900, John Holland, the father of modern submarines, said "…as far as the future that mankind can see, the submarine is a true ocean demon, and relying on it means winning" (Cao et al., 2014). Although the submarine is critical equipment for the Navy to be able to perform hidden missions, it does not prevail in certain environments or against "quiet" threats. For instance, submarines cannot enter coastal areas to directly perform surveillance missions. Its operational effectiveness is thusly greatly affected. Consequently, there is an urgent need to develop new technical means to make up for these shortcomings.
In recent years, the processes of productization and in-dustrialization of underwater robots have led to widespread application in the scientific and military fields. At present, underwater robots are mainly classified into four categories, including AUV (Autonomous Underwater Vehicle), ROV (Remotely Operated Vehicle), UGV (Underwater Glider Vehicle) and HOV (Human Occupied Vehicle) (Allotta et al., 2017;Bachmayer et al., 1998;Gao et al., 2014;Rong, 2008;Liu et al., 2013;Vedachalam et al., 2014). Compared with other types of underwater robots (Xu et al., 2011;Sheng et al., 2013;Gu, 2011), AUVs feature completely autonomous sailing, which greatly reduces the operator's labor intensity. It also features self-carrying energy and it has better mobility. In general, AUVs are the most suitable underwater robots for large-scale investigations, and AUVs have recently been used in mine countermeasure operations, hydrographic surveys, environmental monitoring, and scientific sampling & mapping efforts (Kirkwood, 2008;Nicholson and Healey, 2008;Xu and Li, 2011;Wang et al., 2013). Besides, with the development of AUV and related technologies, AUVs are becoming the best choice for military applications -they are a maritime power multiplier with a wide range of important military uses (Zhou and Zhou, 2011;Chapuis et al., 2002;Chen and Sun, 2011;Watt et al., 2016;Renilson, 2014). And there have been related institutions to study the problems related to submarine recovering the AUV. Recently, the US Bluefin company released a simulation video, recycling a Bluefin-21AUV from the submarine and recycling two Bluefin SandShark Micro-AUVs from the Bluefin-21AUV using a similar method, which is also highly valued by the US Navy and shows that they attach great importance to this technology. At present, the United States uses or proposes four methods for submarine recovery of an AUV, namely, the recitation method, the torpedo launch tube method, the ballistic missile launch tube method, and the docking mode method (Wang, 2013). Compared with surface vessels, recycling by submarine features advantages such as good concealment, small impact of sea weather conditions, long duration of combat, low dependence on other shelter forces, and independent execution of combat missions. AUVs and submarines can work together in complementary fashion to form new combat capabilities and means of warfare. Therefore, the submarine equipped with an underwater unmanned combat platform can continuously update and enrich its function, which is of great significance for improving the comprehensive performance of the submarine. Therefore, AUV deployment/recycling technology development is critical for enabling submarine-equipped AUVs to become widely used (Zhang et al., 2012;Yan et al., 2017). This technique is a problem of two-body fluid dynamics, and relevant research has been conducted on this aspect (Leong, 2014;Cheng, 2006;Zhang, 2008;Leong et al., 2015;Hardy and Barlow, 2008;Gillis, 2014). Cheng (2006) studied the interference problem between the three-dimensional object and the wall surface, which can be used to analyze the mechanical properties when AUV is close to the submarine's surface, but it does not involve related issues with the docking system. Yang (2015) studied the impact of the submarine's turbulence on the AUV's maneuverability and the underwater guidance problem, and he did not consider the problem of the recovery device. However, in the actual recovery process, the mechanical properties of the docking system are decisive for the successful recovery of AUV.
From the above analysis, the most in-depth study of submarine recycling AUV research is the ballistic missile launch tube method, and this method has also completed some applications. However, there is little publicly available information on the recovery of AUVs from submarines, not to mention the publicly available information on the study of the mechanical properties of the recovery devices installed on submarines. This requires a systematic study on the recycling process and method. In view of the above aspects, based on aerial refueling, a cone-shaped recovery system using similar lifting mast method is studied in this paper. Firstly, the numerical analysis method is used to analyze the dynamic characteristics of different shape guiding covers, as well as the influence of the installation position of the recovery system and the length of the rigid rod on the submarine's mechanical properties. Finally, the deflection and vibration problems of the rigid rod are mathematically analyzed.

Mechanical design
In order for a submarine to recycle an AUV, underwater docking of the AUV must first be achieved. Currently, there are three main types of underwater docking devices for AUVs: guided-style, captured-style, and seated-style (Allen et al., 2006;Singh et al., 2001;Kawasaki et al., 2005). For the captured-style docking method, the rope is susceptible to submarine's turbulence and external factors, and special capture mechanisms are required for the AUV's front section. The seated-style docking method can be applied to the static docking, but the actual application is relatively less, and the method requires high maneuverability of the AUV. Therefore, the most researched and mature application is the guided-style docking method. Wu et al. (2014) studied the hydrodynamic analysis of AUV underwater docking with a cone-shaped dock under ocean currents. Zhang et al. (2017) also studied the impact process of AUV underwater docking with a cone-shaped dock. The Kongsberg Company (Circle, 2012) introduced underwater mobile docking of AUVs with a guiding cover. Allen et al. (2006) described the guiding docking system developed for the REMUS AUV. The Bluefin Robotics Company (Circle, 2012) introduced the UUV docking and recharging station, demonstrating the results and suggesting next steps. Park et al. (2009) and Dunbabin et al. (2008) conducted tests and studied the guiding method for underwater AUV docking systems. In addition, there are presently many improvements in the current guided-style docking device. For example, the direction of the guiding cover can be rotated 360° as needed. In addition, it can be seen from the introduction that the current recovery method successfully applied to the submarine is the ballistic missile method. If the submarine uses the ballistic missile tube to recover AUV, the best docking method should be the guided-style. Therefore, the guided-style docking method is next used as a research object of the recovery device, that is, a cone-shaped device.
One must also consider the connection between the docking device and the submarine, which is somewhat similar to aerial refueling. Currently, there are two methods of aerial refueling -the flying boom system and the probeand-drogue system -each with its advantages and disad-vantages (Wu et al., 2016). By comparison, it can be found that the captured-style docking method is similar to the probe-and-drogue system, and the guided-style docking method is similar to the flying boom system. However, there are certain differences between them. First of all, the final task is different. Aerial refueling simply transfers oil from a tanker to a receiver, rather than recycling a receiver to a tanker. Second, the docking velocity is different. The velocity of the submarine and AUV is not in the same order of magnitude as the velocity of the aircraft (Liu et al., 2018). In addition, the physical properties of air and water are also very different, which leads to some differences in the mechanical properties of the aircraft refueling and underwater docking.
To verify the hydrodynamic characteristics of the connection method of a probe-and-drogue system underwater, we used a USV to conduct a preliminary lake test certificate for a flexible rope towing a stable wing, as shown in Fig. 1. The stable wing is connected to the pressure sensor by a short rope, the pressure sensor is connected to the force sensor via a longer rope, and the force sensor is fixed to the USV.
Through experimental tests, it can be seen that the stable wing's depth is related with its velocity. The smaller velocity of the stable wing, the greater sailing depth. Moreover, at the same velocity, the pressure sensor's sailing depth with the stable wing is greater than that without the stable wing, as shown in Fig. 2. And the depth of stable wing changes less with the stable wing as the towling velocity increases, which showed that the stable wing is less affected by towling velocity and this is good for recovering the AUV. However, under different sea conditions, the stable sailing depth of the stable wing will be different, and the rope is also susceptible to the influence of the current, which will introduce certain difficulties when trying to recover the AUV, so the captured-style docking method with rope is not suitable for submarine recovering AUV.
In order to solve this problem when utilizing a submarine, one should use the flying boom method to recover the AUV with a rigid rod, as shown in Fig. 3. This basically guarantees that under different sea conditions, the guiding cover will be stable at a certain fixed depth during the sailing process. At the same time, in order to protect the AUV and recovery system, some soft materials can be installed in the guide housing. When the AUV and the recovery system collide, the soft materials can play a role in buffering. In addition, similar to the flying boom system, the recycling rod preferably has a telescopic function for recycling an AUV.
In the process of submarine recovering AUV, the submarine usually sails counter currently at a slower velocity, and the AUV usually chases the submarine at a certain fixed depth and at a higher velocity (or the submarine hovers at a certain depth while the AUV gradually approaches the submarine). Because when they sail counter currently under the currents, it is easier and more effective to adjust their maneuverability. When the AUV is close to the submarine, the submarine will extend the recovery device to complete the capture and recovery of the AUV. When the submarine is in cruise state, the recovery system is inside the submarine. Because if the submarine sails while the recovery system is extended, the recovery system not only increases the total drag, but also affects the maneuverability of the submarine. Therefore, attention should be paid to the influence of the hydrodynamic characteristics of the recovery system on the submarine's AUV recycling process.
Here we use Suboff as the simulation object because it is a standard submarine scale model recognized by scholars and has experimental data. In addition, the correctness of setting parameters in numerical simulation can be verified by comparison with the experimental data. The total length of the model is L=4.356 m, and the maximum diameter of the main body is D=0.508 m. For the specific geometric   MENG Ling-shuai et al. China Ocean Eng., 2020, Vol. 34, No. 3, P. 387-399 389 model, please refer to Groves et al. (1989). Usually, the outer diameter of the guided recovery unit is four times the AUV's diameter, the length of a conventional submarine is about 80 m, and the length of a light AUV is about 6 meters, so the length ratio is about 13.3. In order to meet the above conditions, the AUV model used in Fig. 3 is a Remus 100AUV with a four-fold reduction in overall size, that is, d=47.75 mm, l=327 mm. The total length of the guiding cover is a=277 mm, the diameter of the front end of the cover is d 1 =62.5 mm, the maximum diameter of the funnel is d 2 =262 mm. In addition, because the guide cover and the submarine are connected by a recycling rod, we also need to determine the properties of the recycling rod. With the actual application scenario, it is better to use a rigid rod. Because the rigid rod has a large bending stiffness, it can effectively resist the interference of external factors. The diameter of the rigid rod is d 3 =40 mm. Besides, since the storage of the ballistic missile is generally upward, in order to use the storage to recover AUV, the recovery system is better to be installed above the submarine. For the guiding cover, there are two forms: closed structure and frame structure, as shown in Fig. 4.
Considering the front, middle, and rear of the submarine, three points (A, B, C) for the installation of the recovery system are set. The distance from Point A to the head end of the Suboff is 850 mm (L 1 ), and the distances between Point A and Point B and Point B and Point C are both 1100 mm (L 2 ). In addition, the distance from the guiding cover to the axis of the submarine is H. In order to verify the influence of the length of the rigid rod on the Suboff, H is set to 400 mm, 600 mm, 800 mm, and 1000 mm respectively, as shown in Fig. 5. Of course, the recovery system can be in-stalled anywhere on the upper surface of the submarine, and this involves optimization of the installation location. Since the focus of this paper is not optimization, only three key installation points have been selected for analysis.

CFD simulation analysis
ε Over the past decade, with the rapid increase of computer speed and capacity and the development of CFD theory and technology, CFD technology has been used more and more widely for various hydrodynamic performance studies (Hu, 2013). The hydrodynamic simulation software used in this paper is STAR-CCM. According to the experience of numerical simulation and verification, this software and turbulence model k− are suitable for outfield flow, widely used in the simulation process of the real project, and the simulation accuracy also meets the engineering requirements (Li and Liu, 2017;Jagadeesh et al., 2009). The final simulation results are shown in Table 1. Among them, the simulation value is the drag of the Suboff when it goes straight at a velocity of 3.045 m/s with the attached body. Table 1 shows that as the number of cells increases or the based size decreases, the simulation error gradually decreases. When the number of cells is 237100, that is, the reference size is 0.45 m, the simulation error is about 1.1%, which satisfies the engineering application requirements, indicating that the mesh division, the numerical turbulence model and the simulation parameters are all possible. Although, as the number of cells continues to increase, the simulation error will still be reduced, the simulation time is greatly increased. In order to improve the simulation efficiency under the premise of ensuring the calculation accuracy, we finally selected the based size of 0.45 m. Therefore, in the following simulation process, the same grid scale and parameter settings are applied. In order to obtain more ac-   1.1% 0.8% Note: Based size is a reference size, and the specific size of the mesh is based on the based size, taking a certain proportion of the size. For example, the based size is 0.45 m, and take the Suboff surfacemesh size to 1% of the based size, i.e. 4.5 mm. ε curate results, we performed a mesh refinement on the recycling system and set an encryption zone around the submarine. Similar to the Suboff model simulation, since the recycling process is an external flow problem around complex rigid geometries, the k− turbulence model and rigidbody are still selected. As shown in Fig. 6, the domain is a cylindrical computational space, its size is Ф9 m×22 m, L 1 =15D, L 2 =20D, d=17.5D, where D is the diameter of the Suboff. In the domain parameter settings, the surface of the Suboff is set to a non-slip condition, the front end of the domain is set as the velocity inlet condition, and the rear end is set as the pressure outlet condition, and the outer surface of the cylindrical watershed is set as the slip surface condition.

Influence of H on submarine hydrodynamics
When the recovery system is installed at Point B, the velocity is 1 m/s, and H is set to varied values and the velo-city distribution of the surrounding fluid around the Suboff model is shown in Fig. 7. For closed structure, as H decreases, the interference between the recovery system and the Soboff will gradually increase. When H>600 mm, the wake generated by the guiding cover does not overlap with  MENG Ling-shuai et al. China Ocean Eng., 2020, Vol. 34, No. 3, P. 387-399 the wake of the Soboff. In addition, as H decreases, the wake behind the guiding cover tends to approach the surface of the Soboff, which indicates that the rigid rod in the recovery device cannot be too short, otherwise the AUV may collide with the Soboff during the recovery process. When H = 400 mm, the wake generated by the guiding cover and the wake of the Soboff have overlapped each other and the hydrodynamic characteristics of the Soboff have been affected. For the frame structure, the wake generated by the guiding cover is small, which is mainly caused by the rigid rod. From Fig. 7a, there is a large area of the reverse velocity zone behind the closed structure. It is known from the fluid theory that this will cause an increase in the differential pressure of the Suboff. Compared with the closed structure, the reverse velocity zone behind the frame structure is smaller in Fig. 7b, so the Suboff with the frame structure guiding cover will have less drag. However, as H decreases, the fluid is between the guiding cover and the Suboff gradually produces a reverse velocity zone.
When the velocity is 1 m/s, the curve of the Suboff model's drag with H is shown in Fig. 8. From Fig. 8a, when the recovery system uses the closed structure and is installed at Points A and B, the Suboff model's drag increases as H increases, which is also in line with the corres-ponding theory of fluid mechanics because the wet surface area of the recovery system increases accordingly. However, when the recovery system is installed at Point C, the Suboff model's drag decreases first and then increases as H increases. In addition, as H increases, the recovery system installed at Points A and B gradually reduces the difference in the drag of the Suboff model. When the velocity takes other values, the length of the rigid rod in the recovery system has similar results for the Suboff model's drag. From Fig. 8b, no matter the recovery system is installed at Point A, B or C, the Suboff model's drag increases as H increases. Besides, when the recovery system is installed at different points, the Suboff's drag change is not obvious. However, it can be seen that in the case of the same H, when the recovery system is installed at Point C, the Suboff has the largest drag.
When the velocity is 2 m/s, the curve of the Suboff model's lift with H is shown in Fig. 9. When the recovery system is installed at Points A and B, the lift value of the Suboff model decreases as H increases, which is exactly the opposite of the drag change trend. This shows that as the length of the rigid rod increases, the interference between the recovery system and the Suboff also decreases. In addition, when H is smaller than 600 mm, the lift value of the  Suboff model varies drastically with the length of the rigid rod, especially when the recovery system is installed at Point B. When the closed structure guiding cover is installed at Point C, the lift value of the Suboff model decreases first and then increases with the increase of H, and finally remains basically unchanged, as shown in Fig. 9a. When the recovery system is installed at Points A and B, and H takes the same value (H≤600 mm), the Suboff with the frame structure has smaller lift value than that with the closed structure. However, as H increases (H>600 mm), at the same velocity and the same H, the Suboff's lift is basically the same no matter which structure guiding cover is used. When the velocity takes other values, the length of the rigid rod of the recovery system has similar results for the lift of the Suboff model. (Note: the negative sign of lift indicates that the submarine's force in the z direction is downward.) When the recovery system uses the closed structure guiding cover and is installed at Point C and the velocity is 1 m/s, the velocity distribution of the surrounding fluid around the Suboff model is shown in Fig. 10. When H= 1000 mm, the wake of the rigid rod interferes greatly with the flow field at the rear of the Suboff, but the wake after the cover does not have a great impact on the Suboff. As H decreases, the interference of the rigid rod to the flow field at the rear of the Suboff is reduced, but the wake behind the cover tends to gradually approach the surface of the Suboff. When H ≥ 600 mm, the wake generated by the cover and the wake of the Soboff do not overlap with each other in a wide range. When H = 400 mm, the wake generated by the cover completely overlaps with the wake of the Suboff, which will have certain influence on the maneuverability and dynamic characteristics of the Suboff.
Since the Suboff model and the recovery system are symmetrical about the longitudinal plane, the force in the ydirection is substantially close to 0.
3.2 Influence of the installation location of the recovery system on hydrodynamics When the recovery system is installed at three different positions A, B, C, the velocity is 1 m/s and H = 600 mm, the pressure distribution of the surrounding fluid around the Suboff model is shown in Fig. 11. When H = 600 mm, the velocity distribution is shown in Fig. 12. When the closed structure guiding cover is installed at Point A, the low-pressure region in the front of the Suboff will partially overlap with the low-pressure region of the cover. When the recovery device is installed at Point C, the low-pressure area behind the cover will partially overlap with the low-pressure area created by the rear of the Suboff. When the recovery device is installed at Point B, the low-pressure zone in front of and behind the cover and the low-pressure zone generated in the front and rear of the Suboff have the least influence on each other. In addition, in the case of the same H and the same velocity, when the recovery device is installed at Point C, the wake behind the cover is closer to the surface of the Suboff. Unlike the closed structure guiding cover, when the recovery system uses the frame structure guiding cover, no matter where the recovery system is installed, the interaction between the recovery system and the submarine is relatively small. However, there is a counterflow zone between the guiding cover and the Suboff, when the recovery device is installed at Point B or C. When the recovery system is installed at three different positions A, B, C, and H = 400 mm, the total drag of the Suboff is shown in Fig. 13. Regardless of where the recovery system is installed on the Suboff, the drag of the Suboff increases with the increasing velocity, which is consistent with the theory of fluid mechanics. For the closed structure guiding cover, when the recovery system is installed at Point C, the drag is the largest; when the recovery system is installed at Point B, it experiences the least drag; when the recovery system is installed at Point A, the drag is slightly Fig. 10. Velocity distribution of the surrounding fluid under different H as the recovery system is installed at Point C (v=1 m/s). larger than that installed at Point B. Moreover, as the velocity increases, the difference between the drag of the recov-ery system installed at Point C and the drag of the system installed at Points A and B is increasing. However, in case of the same velocity and the same H, the Suboff's drag with the frame structure guiding cover is half of the drag with the closed structure guiding cover. And when the recovery system used the frame structure guiding cover, no matter where the recovery system is installed, the total drag of the Suboff is relatively close. In addition, when the recovery system extends out of the Suboff and the Suboff is at a higher velocity, the total drag of the Suboff will increase significantly. This is also why the submarine is usually in a hovering state or a low-velocity state when it recovers the AUV. When H takes other values, the installation position of the recovery system has similar results for the drag of the Suboff.
Through the above analysis, we can see that the installation position of the recovery system, the length of the rigid rod, and the form of the guiding cover all have an influence on the dynamic characteristics of the Suboff. In addition, the recovery system is a convex body for the Suboff, and has a   great influence on its drag during the sailing process, so the recycling process is better as follows: when the Suboff recovers an AUV, the Suboff is preferably in a hovering state or a low-velocity state, and the recovery system is extending out. After the AUV completes the docking process with the guiding cover, the recovery system will be placed inside the Suboff. In other words, the recover system will only extend outside the Suboff during the recycling process. In addition, the influence of the frame structure guiding cover on the hydrodynamic characteristics of the Suboff is smaller than that of the closed structure guiding cover to the Suboff.

Mathematical analysis of the recovery system
In the process of a submarine recycling an AUV underwater, the recovery system will suffer from external disturbances caused by hydrodynamic forces and various other factors. Under the combined influence of external factors, the rigid rod will undergo the deformation and vibration phenomena (Wang et al., 2015). The following section covers the analysis of the rigid rod's deformation and vibration from a mathematical point of view.

Deflection of the rigid rod ω
In engineering, deflection is the degree to which a structural element is displaced under a load, which may refer to an angle or a distance. When the rigid rod is subjected to the change of a force or non-uniform temperature, it will produce deflection. The law of deflection of each point on the rigid rod with position and time is called deflection function or displacement function (z) (Jones, 1971;Yang et al., 2011;Zhu et al., 2006). The rigid rod deflection diagram is shown in Fig. 14. Through simulation and analysis, the force that causes the rigid rod to deform is mainly the drag of the rigid rod itself and the cover. In addition, it can be seen from Section 3 that as the velocity increases or there is a large current, the recovery system's drag increases significantly, especially the cover. And in order to protect the recovery system and underwater robots, the rigid rod can be made of the material with low stiffness. When the collision occurs, the recovery rod plays a certain buffering role. If the deflection of the rigid rod is too large, the orientation of the cover will change, which will affect the success rate of underwater docking. Therefore, the deflection of the recovery rod needs to be analyzed and it must be within a certain reasonable range. (Note: since the rigid rod and the cover are both axisymmetric, the deflection of the rigid rod is mainly in the x-direction, and the deflection in the y-direction and z-direction is essentially zero.) Since the recovery system is connected to the Suboff through one end of the rigid rod, and the cover is in a free state, it is known from the material mechanics that the rigid rod can be treated as a cantilever beam. Through the force analysis of the recovery system, the rigid rod is mainly affected by the Suboff's pulling force, its own drag, and the pulling force of the cover. The bottom end of the rigid rod is treated as the fixed end, the upper end is treated as the free end. The pulling force of the cover on the rigid rod is handled as the concentrated load, and the drag of the rigid rod itself is a uniform load. According to theoretical mechanics, the deflections generated by these two forces can be superimposed on each other.
where, EI y denotes the bending stiffness of the rigid rod, and M(z) denotes the bending moment of the rigid rod. When only the rigid rod's drag acts on itself, the bending moment of the rigid rod is: where, q denotes the drag of the rigid rod unit length, and l is the length of the rigid rod. Considering the boundary conditions: We obtain: When only the pulling force of the cover acts on the rigid rod, the deflection of the rigid rod is: where, D cx is the drag of the guiding cover. By applying the principle of superposition, the total deflection of the rigid rod is: Generally, the maximum normal stress occurs at the front and rear edges of the rigid rod section. In addition, information such as the strength of the rigid rod can be checked by the normal bending stress formula and shear stress formula. From Fig. 14, Eq. (3) and Eq. (4), we know that the maximum deflection of the rigid rod occurs at the junction of the guiding cover and the rigid rod. In order to minimize the deflection of the rigid rod, the length of the rigid rod should be minimized, or the rigid rod with high bending stiffness should be used. In addition, it is also possible to reduce the deflection by optimizing the shape of the guiding cover or the recovery rod to reduce its own drag.
According to the above analysis, the rigid rod will be deformed during the movement. Through numerical simulation, we can put the drag of the recovery system into Eq. (5) to obtain the deflection of the rigid rod. Generally, since the deformation is not so large, the force change of the rigid rod is not obvious. Therefore, when we use the deformation model of the recovery device, the influence of deformation on hydrodynamics is not considered.

Transverse vibration of the rigid rod
Because the rigid rod has certain rigidity, the rigid rod can play a certain protective role when the AUV and the recovery device have relative motion. However, since the recovery system is subjected to a large force in the x direction, the rigid rod may undergo a certain lateral vibration in the presence of interference from the outside.
Referring to Fig. 15, we consider the rod stretched between two points at z=0 and z=l and displaced by a distribution of external force. Conservation of transverse momentum requires that the total lateral force on the rod element be balanced by its inertia. Let the lateral displacement be V(z, t) and consider a differential element between z and z+dz. The net transverse force due to the difference of tension at both ends of the element is where T denotes the local tension in the rigid rod and ∂V ∂z << 1 α We shall assume the lateral displacement to be small everywhere so that the slope is also small: . The local value of sin can then be determined by δ δ where the expression O( ) stands for the order of .
According to the Taylor formula, the net lateral force is According to Hooke's law, we see that the initial tension is where E denotes the Young's modulus of rigidity and S denotes the cross-sectional area of the rigid rod. For simplicity, E and S, and hence T, will be assumed to be uniform in z. With lateral displacement, the strain must be changed. Consider the part of the rod extending from 0 to z. The length l (z, t) of this deformed part is: Hence the corresponding strain is: T ∂ 2 V ∂z 2 dz which is of the second-order smallness. As long as (l−z)/z=O[(l−L/L)]<<ΔL/L, the tension is essentially unchanged (Meng et al., 2018). i.e., T is indistinguishable from its initial constant value. Thus, the net lateral force on the rod element is well represented by If the mass per unit length of the rigid rod is , the inertia of the element is . Let the applied load per unit length be q(z, t). By balancing force and inertia, we obtain: ρdz Eliminating dz and taking the limit of dz→0, we obtain: ρ This equation, called the wave equation, is a partial differential equation of the second order. In Eq. (14) the highest derivatives with respect to z and t are of second order, hence we need two boundary conditions, one at z=0 and one at z=l, and two initial conditions at t=0. In addition, through the above analysis, it can be known that the lateral vibration of the rigid rod also has a certain relationship with its own deflection.
Longitudinal displacement caused by lateral forces Is the longitudinal displacement U along the z direction important in this problem? Conservation of momentum in the z direction requires that:  Since: The acceleration is of the second-order smallness: .
Thus, the longitudinal motion is negotiable in comparison.

Longitude vibration of the rigid rod
It can be seen from Fig. 8 that when the recovery system is installed at Point A or B, the lift value in the z-direction will gradually increase as H decreases, which will cause deformation and vibration of the rigid rod. The following describes the longitudinal vibration of the rigid rod.
Consider a rigid rod with the cross-sectional area S(x) and Young's modulus E, as shown in Fig. 16.
Let the longitudinal displacement from equilibrium be U (z, t). The strain at the station z is: (18) By Hooke's law, the tension at z is:

∂U ∂z
Now the net tension on a rod element from z to z +dz is: Let the externally applied longitudinal force be F(z, t) per unit length. Momentum conservation requires that: In the limit of vanishing dz, we obtain the differential equation: In the special case of uniform cross section, S=constant, and U satisfies the inhomogeneous wave equation: has the dimension of velocity. For the recovery system, the lower end is fixed but the upper end is free, which gives boundary conditions: Since the stress is proportional to the strain. Again, the most natural initial conditions are: , and where f and g are prescribed functions of x for 0<z<H.
With the heavy mass M of the cone-shaped cover attached to the end z=H, momentum conservation of the mass M requires that: This equation serves as a boundary condition for the rigid rod, now involving both and at the end.

Conclusions
By referring to the flying boom aerial refueling system, a cone-shaped recovery system with two different guiding covers (closed structure and frame structure) for submarine recycling an AUV is introduced. With the Suboff model as the research object, the hydrodynamic analysis of the recovery system is carried out, including the hydrodynamic influence of the recovery system on the Suboff model when the recovery system is installed at different positions and the rigid rod is of varied lengths. Finally, the mathematical model analysis of the deformation and vibration (transverse and longitudinal) of the rigid rod is carried out. Through simulation analysis, it is found that: (1) In the case of the same velocity and the same length H of the rigid rod, the submarine's drag is the largest when the recovery system is installed in the front of the Suboff. When the recovery system is installed in the middle and rear of the Suboff model, the Suboff model has the good stability and less drag.
(2) For the recovery system, the frame structure guiding cover is better than the closed structure guiding cover for the submarine recovering an AUV, because the frame structure has less drag and better stability at the same velocity and installation position.
(3) Regardless of the installation location of the recycling system, the Suboff's drag increases as the length of the rigid rod increases, except when the closed structure guiding cover is installed at the rear of the Suboff. When the recovery system uses the closed structure guiding cover and is installed at the rear of the Suboff, the wake behind the cover tends to gradually approach the surface of the Suboff, and the drag decreases first and then increases with the increase of the length of the elastic rod.
(4) Generally speaking, the lift value decreases as the length of the rigid rod increases. When the recovery system is installed in the front and middle of the Suboff model, and H takes the same value (H ≤ 600 mm), the Suboff with the frame structure has smaller lift value than that with the closed structure. However, as H increases (H>600 mm), at the same velocity and the same H, the Suboff's lift is basically the same no matter which structure guiding cover is used.
(5) In general, the recovery system is best installed in the middle of the submarine because the total drag is the smallest and the impact of the recovery system on the submarine is minimal. Besides, the maximum deflection of the rigid rod occurs at the junction of the guiding cover and the rigid rod. In order to minimize the deflection of the rigid rod, in the case of a reasonable length of the rigid rod, the rigid rod should use a material with high bending stiffness and the shape of the recovery system should be optimized.
As for the vibration equation of the rigid rod, it is partial differential equation of the second order and is related to the force, length and external disturbance. Besides, It is not easy to simulate using CFD software. In the future, we will make some tests of the vibration and the collision of the AUV with the recovery system.